# Are there 'complex' numbers in classical mechanics?

1. Jul 23, 2011

### lmerriam

Wondering if it is only the formulae of quantum mechanics that routinely include complex numbers (a real component plus an imaginary one, e.g. i (the square root of -1)). If so, doesn't this immediately suggest (or even demand) that the (un)reality of the quantum realm is fundamentally unlike the classical one we experience?
thanks!

2. Jul 23, 2011

### SpectraCat

Any oscillating function (e.g. EM radiation) can be described by a complex exponential. Recall the Euler formula: $e^{\pm ix}=cos x \pm i sin x$

So no, complex numbers are not unique to quantum mechanics.

Last edited: Jul 23, 2011
3. Jul 23, 2011

### unusualname

In QM we require complex numbers at fundamental level to account for phases of probability amplitudes.

In Classical Mechanics complex numbers are hugely useful in wave equations etc, but are not required fundamentally, the equations are always separable (in principle) into real-valued parts, and the observables are real-valued functions.

4. Jul 23, 2011

### saim_

Quantum notions ARE fundamentally different from the classical notions we had before quantum theory. But, even in quantum mechanics, normally, observables are represented by Hermitian operators with real eigenvalues so complex numbers (mostly) don't show up as observable values of measurable quantities; just like in classical mechanics, consistent with observation. On the other hand, impedance is clearly complex and was present before quantum mechanics. Similarly the wave equations. So taking into account all the weirdness of quantum mechanics, I don't think presence of a lot of complex algebra is something very deep and meaningful by itself.

5. Jul 23, 2011

### unusualname

How is impedance "clearly complex"?

The only thing "clearly complex" in nature is QM probability amplitudes, and the OP is right to point out that that this differentiates QM from the classical.

6. Jul 23, 2011

### saim_

@unusualname: Electrical impedance is not clearly complex?

7. Jul 23, 2011

### unusualname

It's constructed to be so, but to measure it you calculate real-valued quantities. Impedance is not a fundamentally complex-valued property of nature.

8. Jul 23, 2011

### unusualname

To be more precise, in classical mechanics you can always decouple the complex valued equations into real and imaginary parts (but the calculations may become incredibly difficult)

But in QM you cannot, this is not discussed so much in modern books (it's obvious now?) but for example see Bohm's classic Quantum Theory book p84:

9. Jul 23, 2011

### mathfeel

It is just convenient to write impedance as complex number. Its absolute value gives the ratio between amplitude of AC voltage and current. Its complex phase gives the phase shift between the two quantity. You can still decouple them and use sine and cosine with phase shift.

It is however easier to do certain calculation in classical physics in complex number, but it's not fundamental.

10. Jul 23, 2011

### saim_

@unusualname: I'm not trying to show QM is the same as CM or that complex numbers have the same status in CM as QM. I'm just saying they are not that voodoo in physics without QM as well and I disagree that their presence is the reason of fundamental difference between the "reality of QM and CM". You should try to understand the point of a post before commenting.

11. Jul 23, 2011

### unusualname

Well sorry, but the reason for complex numbers in QM is really VERY different to the reason for complex numbers in classical mechanics, and since that was the point of the OP's post I hope you agree that you were corrected fairly.

QM requires complex numbers, it requires an amplitude and phase for probability amplitudes, anything else you might come up with will be equivalent.

12. Jul 23, 2011

### saim_

Yes, OK. But do you agree with this:
Probability amplitudes are complex; nobody is arguing with that. But is that the reason QM is different from CM? The fundamental reason that makes QM different from CM? This is what I was saying no to in my post. Just because something has complex quantities to be dealt with doesn't make it all that much like QM, does it?

13. Jul 23, 2011

### unusualname

Well, if you understand how the complex quantities are introduced then it does make a difference. I already linked to a classic textbook available on google books which discusses this.

There is no "weirdness", there is simply a complex probabilty amplitude in nature, which doesn't exist in classical mechanics.

14. Jul 23, 2011

### Hurkyl

Staff Emeritus
Anything involving complex numbers can always be split into entirely real parts. That line of thinking is a red herring and not useful.

If properties of complex numbers (e.g. addition and multiplication) corresponds to some relevant aspect of what is being quantified, then that's generally how you should think of things -- it would be counter-productive to try and force yourself to mentally separate the complex numbers into a pair of real numbers.

15. Jul 23, 2011

### akhmeteli

This is certainly a popular opinion, but is it correct? See my post https://www.physicsforums.com/showpost.php?p=830490&postcount=9 , quoting Schroedinger, who argued that real numbers can be enough, at least for the Klein-Gordon-Maxwell electrodynamics. Surprisingly, the same is true for the Dirac equation and spinor electrodynamics ( http://arxiv.org/abs/1008.4828 (accepted for publication in the Journal of Mathematical Physics); you can find a somewhat more complete version at http://www.akhmeteli.org/jmp.pdf ).

You quote the Bohm's book, but there is a caveat there: "in the non-relativistic domain, at least", however Nature is not aware of such domain, and relativistic theories are definitely preferable.

Last edited by a moderator: Apr 26, 2017
16. Jul 23, 2011

### kof9595995

You can use complex numbers for classical mechanics , or use real numbers for quantum mechanics, but at the end of the day you'll get theories "isomorphic"(having the same structure) to the conventional ones, so the important point is using which numbers are more efficient, like Hurkyl said.

17. Jul 24, 2011

### akhmeteli

Hurkyl discusses replacement of complex numbers by pairs of real numbers, as far as I understand, and I fully agree that such replacement, while always possible, is not meaningful. However, this is not relevant to the theories I mentioned in post 15 in this thread. For example, a complex wavefunction of the Klein-Gordon equation is replaced by one real wavefunction, and the complex spinor wavefunction of the Dirac equation (i.e. 4 complex functions) is replaced by one real wavefunction. While these theories are pretty much physically equivalent to the traditional ones, I am not sure they have the same structure: on the one hand, they have fewer variables, on the other hand, they are not gauge invariant (the gauge is fixed).

18. Jul 25, 2011

### agentredlum

Something that blew my mind when i learned about it, materials that allow light to be transmitted through them can have an index of refraction that is a complex number. Also look up ordinary ray, extraordinary ray and birefringence. Optics is fascinating!